What is MHD?

Magnetic fields influence many natural and man-made flows. They are routinely used in industry to heat, pump, stir and levitate liquid metals. There is the terrestrial magnetic field which is maintained by fluid motion in the earth's core, the solar magnetic field which generates sunspots and solar flares, and the galactic magnetic field which is thought to influence the formation of stars from interstellar clouds. The study of these flows is called magnetohydrodynamics (MHD). Formally, MHD is concerned with the mutual interaction of fluid flow and magnetic fields. The fluids in question must be electrically conducting and non-magnetic, which limits us to liquid metals, hot ionised gases (plasmas) and strong electrolytes.

The mutual interaction of a magnetic field, B, and a velocity field, u, arises partially as a result of the laws of Faraday and Ampère, and partially because of the Lorentz force experienced by a current-carrying body. The exact form of this interaction is analysed in detail in the following chapters, but perhaps it is worth stating now, without any form of proof, the nature of this coupling.

The History of Electrometallurgy

There were two revolutions in the application of electricity to industrial metallurgy. The first, which occurred towards the end of the nineteenth century, was a direct consequence of Faraday's discoveries. The second took place around eighty years later. We start with Faraday.

The discovery of electromagnetic induction revolutionised almost all of 19th century industry, and none more so than the metallurgical industries. Until 1854, aluminium could be produced from alumina only in small batches by various chemical means. The arrival of the dynamo transformed everything, sweeping aside those inefficient, chemical processes. At last it was possible to produce aluminium continuously by electrolysis. Robert Bunsen (he of the ‘burner’ fame) was the first to experiment with this method in 1854. By the 1880s the technique had been refined into a process which is little changed today (Figure I.1).

In the steel industry, electric furnaces for melting and alloying iron began to appear around 1900. There were two types: arc-furnaces and induction furnaces (Figure 1.2).

Prefaces are rarely inspiring and, one suspects, seldom read. They generally consist of a dry, factual account of the content of the book, its intended readership and the names of those who assisted in its preparation. There are, of course, exceptions, of which Den Hartog's preface to a text on mechanics is amongst the wittiest. Musing whimsically on the futility of prefaces in general, and on the inevitable demise of those who, like Heaviside, use them to settle old scores, Den Hartog's preface contains barely a single relevant fact. Only in the final paragraph does he touch on more conventional matters with the observation that he has ‘placed no deliberate errors in the book, but he has lived long enough to be quite familiar with his own imperfections’.

We, for our part, shall stay with a more conventional format. This work is more of a text than a monograph. Part A (the larger part of the book) is intended to serve as an introductory text for (advanced) undergraduate and post-graduate students in physics, applied mathematics and engineering. Part B, on the other hand, is more of a research monograph and we hope that it will serve as a useful reference for professional researchers in industry and academia. We have at all times attempted to use the appropriate level of mathematics required to expose the underlying phenomena. Too much mathematics can, in our opinion, obscure the interesting physics and needlessly frighten the student.

When an electric current is made to pass through a liquid-metal pool it causes the metal to pinch in on itself. That is to say, like-signed currents attract one another, and so each part of the pool is attracted to every other part. When the current is perfectly uniform, the only effect is to pressurise the liquid. However, often the current is non-uniform; for example, it may spread radially outwards from a small electrode placed at the surface of the pool. In such cases the radial pinch force will also be non-uniform, being largest at places where the current density is highest (near the electrode). The (irrotational) pressure force, – ∇p, is then unable to balance the (rotational) Lorentz force. Motion results, with the fluid flowing inward in regions of high current density and returning through regions of small current.

Perhaps the first systematic experimental investigation of the ‘pinch effect’ in current-carrying melts was that of E F Northrup who, in 1907, injected current into pools of mercury held in a variety of different configurations. It should be noted, however, that industrial metallurgists have been routinely passing large currents through liquid metals since 1886, when Hall and Hèroult first developed the aluminium reduction cell and von Siemens designed the first electric-arc furnace. One of the many descendants of the electric-arc furnace is vacuum-arc remelting (VAR).

We have seen that the relative movement of a conducting body and a magnetic field can lead to the dissipation of energy. This has been used by engineers for over a century to dampen unwanted motion. Indeed, as far back as 1873 we find Maxwell noting: ‘A metallic circuit, called a damper, is sometimes placed near a magnet for the express purpose of damping or deadening its vibrations.’ Maxwell was talking about a magnetic field moving through a stationary conductor. We are interested in a moving conductor in a stationary field, but of course, this is really the same thing. We have already touched upon magnetic damping in Chapter 5, and we discussed some of its consequences in Chapter 6. In particular, we saw that the intense magnetic field in a sunspot locally deadens the convective motions in the outer layer of the sun, thus cooling the spot and giving it a dark appearance. Here we make the jump from sunspots to steelmaking, and describe how magnetic fields are used in certain casting operations to suppress unwanted motion.

There has been a myriad of papers on this topic and at times one is reminded of Rayleigh's indigestion. Here we focus on the unifying themes.

Turbulence is not an easy subject. Our understanding of it is limited, and those bits we do understand are arrived at through detailed and difficult calculation. G K Batchelor gave some hint of the difficulties when, in 1953, he wrote:

Little has changed since 1953. Nevertheless, it is hard to avoid the subject of turbulence in MHD, since the Reynolds number, even in metallurgical MHD, is invariably very high. So at some point we simply have to bite the bullet and do what we can. This chapter is intended as an introduction to the subject, providing a springboard for those who wish to take it up seriously. In order not to demotivate the novice, we have tried to keep the mathematical difficulties to a minimum. Consequently, only schematic outlines are given of certain standard derivations and proofs. For example, deriving the standard form for second- and third-order velocity correlation tensors in isotropic turbulence can be hard work. Such derivations are well documented elsewhere and so there seems little point in giving a blow-by-blow description here.

The amount of energy required to reduce alumina to aluminium in electrolysis cells is staggering. In North America, for example, around 2% of all generated electricity is used to produce aluminium. Worldwide, around 2×1010kg of aluminium are produced annually, and this requires in excess of 1011 kWh p.a. The corresponding electricity bill is around £1010 p.a.! Yet much of this energy (around one half) is wasted in the form of I2R heating of the electrolyte used to dissolve the alumina. Needless to say, strenuous efforts have been made to reduce these losses, mostly centred around minimising the volume of electrolyte. However, the aluminium industry is faced with a fundamental problem. When the volume of electrolyte is reduced below some critical threshold, the reduction cell becomes unstable. It is this instability, which is driven by MHD forces, which is the subject of this chapter.

Interfacial Waves in Aluminium Reduction Cells

Early attempts to produce aluminium by electrolysis

It is not an easy matter to produce aluminium from mineral deposits. The first serious attempt to isolate elemental aluminium was that of Humphrey Davy, Faraday's mentor at the Royal Institution. (In fact, Davy's preferred spelling – aluminum – is still used today in North America.) In 1809 he passed an electric current through fused compounds of aluminium and into a substrate of iron.

A high-frequency induction coil can be used to heat, levitate and stir liquid metal. This has given rise to a number of metallurgical processes, some old (such as induction furnaces) and some new. In this chapter, we shall discuss five.

1. (i) Induction furnaces. These have remained virtually unchanged for the best part of a century, yet we are still unable to calculate reliably the stirring velocity within a furnace!

2. (ii) Cold crucible melting. This is an ingenious process which combines the functions of an induction melter and a continuous caster, all in one device.

3. (iii) Levitation melting. This is now routinely used in the laboratory to melt small specimens of highly reactive metals. Unfortunately, if the levitated drop becomes too large, it tends to drip.

4. (iv) The electromagnetic valve. This provides a non-contact means of modulating and shaping a liquid-metal jet. It is a sort of levitation melter in which the metal is allowed to leak out of the bottom.

5. (v) Electromagnetic casting. Some aluminium producers have replaced the casting mould in a continuous caster by a high-frequency induction coil. Thus, the melt pool is supported by magnetic pressure rather than by mechanical means. It is extraordinary that large ingots, which may be a metre wide and ten metres long, can be formed by pouring the liquid metal into free space and soaking it with water jets!

This chapter is a short introduction to the use of models in atmospheric research and forecasting. In Section 8.1 we explain how a hierarchy of models – simple, intermediate and complex – can be used for gaining understanding of atmospheric behaviour and interpreting atmospheric data. In Section 8.2 we give brief details of the numerical methods used in the more complex theoretical models, while in Section 8.3 we outline the use of these models for forecasting and other purposes. In Section 8.4 we describe an example of a class of laboratory models of the atmosphere. Finally, in Section 8.5, we give some examples of atmospheric phenomena that arise from interactions between basic physical processes and that can be elucidated only with the aid of models of intermediate complexity.

The hierarchy of models

The basic philosophy of atmospheric modelling was outlined in Section 1.2. It was mentioned there that a hierarchy of models, from simple to complex, must be used for understanding and predicting atmospheric behaviour; this hierarchy is illustrated in Figure 8.1. The simple models (‘back-of-the-envelope’ or ‘toy’ models) involve a minimum number of physical components and are described by straightforward mathematical equations that can usually be solved analytically. These models provide basic physical intuition: most of the models considered earlier in this book are of this type. The intermediate models involve a small number of physical components but usually require a computer for solution of the mathematical equations.

In this chapter we show how basic thermodynamic concepts can be applied to the atmosphere. We first note in Section 2.1 that the atmosphere behaves as an ideal gas. Some basic information on the various gases comprising the atmosphere is presented in Section 2.2. The fact that the atmosphere is fairly close to being in hydrostatic balance is used in Section 2.3 to develop some very simple ideas about the vertical structure of the atmosphere. An important quantity related to entropy, the potential temperature, is discussed in Section 2.4. The concept of an air parcel is introduced in Section 2.5 and is used to develop ideas about atmospheric stability and buoyancy oscillations. A brief introduction to the concept of available potential energy is given in Section 2.6.

The rest of the chapter is devoted to the implications of water vapour in the air. Section 2.7 recalls the basic thermodynamics of phase changes and introduces several measures of atmospheric water vapour content. These ideas are exploited in Section 2.8, in which some effects of the release of latent heat are investigated in a calculation of the saturated adiabatic lapse rate, which gives information on the stability of a moist atmosphere. The tephigram, a graphical method of representing the vertical structure of temperature and moisture and calculating useful physical results, is introduced in Section 2.9. Finally, some of the basic physics of the formation of cloud droplets by condensation of water vapour is considered in Section 2.10.